THE GEOMETRY OF ALGEBRAIC STACKS

0DQR

Contents

1. Introduction 12. Versal rings 13. Multiplicities of components of algebraic stacks 54. Formal branches and multiplicities 75. Dimension theory of algebraic stacks 96. The dimension of the local ring 217. Other chapters 23References 25

1. Introduction

0DQS This chapter discusses a few geometric properties of algebraic stacks. The initialversions of Sections 3 and 5 were written by Matthew Emerton and Toby Gee andcan be found in their original form in [EG17].

2. Versal rings

0DQT In this section we elucidate the relationship between deformation rings and localrings on algebraic stacks of finite type over a locally Noetherian base.

Situation 2.1.0DQU Here X is an algebraic stack locally of finite type over a locallyNoetherian scheme S.

Here is the definition.

Definition 2.2.0DQV In Situation 2.1 let x0 : Spec(k) → X be a morphism, where kis a finite type field over S. A versal ring to X at x0 is a complete Noetherianlocal S-algebra A with residue field k such that there exists a versal formal object(A, ξn, fn) as in Artin’s Axioms, Definition 12.1 with ξ1 ∼= x0 (a 2-isomorphism).

We want to prove that versal rings exist and are unique up to smooth factors. Todo this, we will use the predeformation categories of Artin’s Axioms, Section 3.These are always deformation categories in our situation.

Lemma 2.3.0DQW In Situation 2.1 let x0 : Spec(k)→ X be a morphism, where k is afinite type field over S. Then FX ,k,x0 is a deformation category and TFX ,k,x0 andInf(FX ,k,x0) are finite dimensional k-vector spaces.

This is a chapter of the Stacks Project, version a577f147, compiled on Sep 05, 2020.1

THE GEOMETRY OF ALGEBRAIC STACKS 2

Proof. Choose an affine open Spec(Λ) ⊂ S such that Spec(k)→ S factors throughit. By Artin’s Axioms, Section 3 we obtain a predeformation category FX ,k,x0

over the category CΛ. (As pointed out in locus citatus this category only dependson the morphism Spec(k) → S and not on the choice of Λ.) By Artin’s Axioms,Lemmas 6.1 and 5.2 FX ,k,x0 is actually a deformation category. By Artin’s Axioms,Lemma 8.1 we find that TFX ,k,x0 and Inf(FX ,k,x0) are finite dimensional k-vectorspaces.

Lemma 2.4.0DQX In Situation 2.1 let x0 : Spec(k)→ X be a morphism, where k is afinite type field over S. Then a versal ring to X at x0 exists. Given a pair A, A′of these, then A ∼= A′[[t1, . . . , tr]] or A′ ∼= A[[t1, . . . , tr]] as S-algebras for some r.

Proof. By Lemma 2.3 and Formal Deformation Theory, Lemma 13.4 (note that theassumptions of this lemma hold by Formal Deformation Theory, Lemmas 16.6 andDefinition 16.8). By the uniquness result of Formal Deformation Theory, Lemma14.5 there exists a “minimal” versal ring A of X at x0 such that any other versalring of X at x0 is isomorphic to A[[t1, . . . , tr]] for some r. This clearly implies thesecond statement.

Lemma 2.5.0DQY In Situation 2.1 let x0 : Spec(k) → X be a morphism, where kis a finite type field over S. Let l/k be a finite extension of fields and denotexl,0 : Spec(l) → X the induced morphism. Given a versal ring A to X at x0 thereexists a versal ring A′ to X at xl,0 such that there is a S-algebra map A → A′

which induces the given field extension l/k and is formally smooth in the mA′-adictopology.

Proof. Follows immediately from Artin’s Axioms, Lemma 7.1 and Formal Defor-mation Theory, Lemma 29.6. (We also use that X satisfies (RS) by Artin’s Axioms,Lemma 5.2.)

Lemma 2.6.0DQZ In Situation 2.1 let x : U → X be a morphism where U is a schemelocally of finite type over S. Let u0 ∈ U be a finite type point. Set k = κ(u0) anddenote x0 : Spec(k)→ X the induced map. The following are equivalent

(1) x is versal at u0 (Artin’s Axioms, Definition 12.2),(2) x : FU,k,u0 → FX ,k,x0 is smooth,(3) the formal object associated to x|Spec(O∧

U,u0) is versal, and

(4) there is an open neighbourhood U ′ ⊂ U of x such that x|U ′ : U ′ → X issmooth.

Moreover, in this case the completion O∧U,u0is a versal ring to X at x0.

Proof. Since U → S is locally of finite type (as a composition of such morphisms),we see that Spec(k) → S is of finite type (again as a composition). Thus thestatement makes sense. The equivalence of (1) and (2) is the definition of x beingversal at u0. The equivalence of (1) and (3) is Artin’s Axioms, Lemma 12.3. Thus(1), (2), and (3) are equivalent.If x|U ′ is smooth, then the functor x : FU,k,u0 → FX ,k,x0 is smooth by Artin’sAxioms, Lemma 3.2. Thus (4) implies (1), (2), and (3). For the converse, assumex is versal at u0. Choose a surjective smooth morphism y : V → X where Vis a scheme. Set Z = V ×X U and pick a finite type point z0 ∈ |Z| lying overu0 (this is possible by Morphisms of Spaces, Lemma 25.5). By Artin’s Axioms,Lemma 12.6 the morphism Z → V is smooth at z0. By definition we can find an

THE GEOMETRY OF ALGEBRAIC STACKS 3

open neighbourhood W ⊂ Z of z0 such that W → V is smooth. Since Z → U isopen, let U ′ ⊂ U be the image of W . Then we see that U ′ → X is smooth by ourdefinition of smooth morphisms of stacks.The final statement follows from the definitions as O∧U,u0

prorepresents FU,k,u0 .

Lemma 2.7.0DZS In Situation 2.1. Let x0 : Spec(k) → X be a morphism such thatSpec(k)→ S is of finite type with image s. Let A be a versal ring to X at x0. Thefollowing are equivalent

(1) x0 is in the smooth locus of X → S (Morphisms of Stacks, Lemma 33.6),(2) OS,s → A is formally smooth in the mA-adic topology, and(3) FX ,k,x0 is unobstructed.

Proof. The equivalence of (2) and (3) follows immediately from Formal Deforma-tion Theory, Lemma 9.4.Note that OS,s → A is formally smooth in the mA-adic topology if and only ifOS,s → A′ = A[[t1, . . . , tr]] is formally smooth in the mA′ -adic topology. Hence (2)does not depend on the choice of our versal ring by Lemma 2.4. Next, let l/k bea finite extension and choose A → A′ as in Lemma 2.5. If OS,s → A is formallysmooth in the mA-adic topology, then OS,s → A′ is formally smooth in the mA′ -adictopology, see More on Algebra, Lemma 36.7. Conversely, if OS,s → A′ is formallysmooth in the mA′ -adic topology, then O∧S,s → A′ and A → A′ are regular (Moreon Algebra, Proposition 48.2) and hence O∧S,s → A is regular (More on Algebra,Lemma 40.7), hence OS,s → A is formally smooth in the mA-adic topology (samelemma as before). Thus the equivalence of (2) and (1) holds for k and x0 if andonly if it holds for l and x0,l.Choose a scheme U and a smooth morphism U → X such that Spec(k) ×X U isnonempty. Choose a finite extension l/k and a point w0 : Spec(l)→ Spec(k)×X U .Let u0 ∈ U be the image of w0. We may apply the above to l/k and to l/κ(u0)to see that we can reduce to u0. Thus we may assume A = O∧U,u0

, see Lemma2.6. Observe that x0 is in the smooth locus of X → S if and only if u0 is in thesmooth locus of U → S, see for example Morphisms of Stacks, Lemma 33.6. Thusthe equivalence of (1) and (2) follows from More on Algebra, Lemma 37.6.

We recall a consequence of Artin approximation.Lemma 2.8.0DR0 In Situation 2.1. Let x0 : Spec(k) → X be a morphism such thatSpec(k) → S is of finite type with image s. Let A be a versal ring to X at x0. IfOS,s is a G-ring, then we may find a smooth morphism U → X whose source is ascheme and a point u0 ∈ U with residue field k, such that

(1) Spec(k)→ U → X coincides with the given morphism x0,(2) there is an isomorphism O∧U,u0

∼= A.Proof. Let (ξn, fn) be the versal formal object over A. By Artin’s Axioms, Lemma9.5 we know that ξ = (A, ξn, fn) is effective. By assumption X is locally of finitepresentation over S (use Morphisms of Stacks, Lemma 27.5), and hence limit pre-serving by Limits of Stacks, Proposition 3.8. Thus Artin approximation as in Artin’sAxioms, Lemma 12.7 shows that we may find a morphism U → X with source afinite type S-scheme, containing a point u0 ∈ U of residue field k satisfying (1) and(2) such that U → X is versal at u0. By Lemma 2.6 after shrinking U we mayassume U → X is smooth.

THE GEOMETRY OF ALGEBRAIC STACKS 4

Remark 2.9 (Upgrading versal rings).0DR1 In Situation 2.1 let x0 : Spec(k) → X bea morphism, where k is a finite type field over S. Let A be a versal ring to X atx0. By Artin’s Axioms, Lemma 9.5 our versal formal object in fact comes from amorphism

Spec(A) −→ Xover S. Moreover, the results above each can be upgraded to be compatible withthis morphism. Here is a list:

(1) in Lemma 2.4 the isomorphism A ∼= A′[[t1, . . . , tr]] or A′ ∼= A[[t1, . . . , tr]]may be chosen compatible with these morphisms,

(2) in Lemma 2.5 the homomorphism A→ A′ may be chosen compatible withthese morphisms,

(3) in Lemma 2.6 the morphism Spec(O∧U,u0) → X is the composition of the

canonical map Spec(O∧U,u0)→ U and the given map U → X ,

(4) in Lemma 2.8 the isomorphism O∧U,u0∼= A may be chosen so Spec(A)→ X

corresponds to the canonical map in the item above.In each case the statement follows from the fact that our maps are compatiblewith versal formal elements; we note however that the implied diagrams are 2-commutative only up to a (noncanonical) choice of a 2-arrow. Still, this means thatthe implied map A′ → A or A → A′ in (1) is well defined up to formal homotopy,see Formal Deformation Theory, Lemma 28.3.

Lemma 2.10.0DR2 In Situation 2.1 let x0 : Spec(k) → X be a morphism, where k isa finite type field over S. Let A be a versal ring to X at x0. Then the morphismSpec(A)→ X of Remark 2.9 is flat.

Proof. If the local ring of S at the image point is a G-ring, then this followsimmediately from Lemma 2.8 and the fact that the map from a Noetherian localring to its completion is flat. In general we prove it as follows.Step I. If A and A′ are two versal rings to X at x0, then the result is true for Aif and only if it is true for A′. Namely, after possible swapping A and A′, we mayassume there is a formally smooth map ϕ : A→ A′ such that the composition

Spec(A′)→ Spec(A)→ Xis the morphism Spec(A′)→ X , see Lemma 2.4 and Remark 2.9. Since A→ A′ isfaithfully flat we obtain the equivalence from Morphisms of Stacks, Lemmas 25.2and 25.5.Step II. Let l/k be a finite extension of fields. Let xl,0 : Spec(l)→ X be the inducedmorphism. Let A be a versal ring to X at x0 and let A→ A′ be as in Lemma 2.5.Then again the composition

Spec(A′)→ Spec(A)→ Xis the morphism Spec(A′)→ X , see Remark 2.9. Arguing as before and using stepI to see choice of versal rings is irrelevant, we see that the lemma holds for x0 ifand only if it holds for xl,0.Step III. Choose a scheme U and a surjective smooth morphism U → X . Then wecan choose a finite type point z0 on Z = U ×X x0 (this is a nonempty algebraicspace). Let u0 ∈ U be the image of z0 in U . Choose a scheme and a surjective étalemap W → Z such that z0 is the image of a closed point w0 ∈ W (see Morphisms

THE GEOMETRY OF ALGEBRAIC STACKS 5

of Spaces, Section 25). Since W → Spec(k) and W → U are of finite type, wesee that κ(w0)/k and κ(w0)/κ(u0) are finite extensions of fields (see Morphisms,Section 16). Applying Step II twice we may replace x0 by u0 → U → X . Then wesee our morphism is the composition

Spec(O∧U,u0)→ U → X

The first arrow is flat because completion of Noetherian local rings are flat (Algebra,Lemma 96.2) and the second arrow is flat as a smooth morphism is flat. Thecomposition is flat as composition preserves flatness.

Remark 2.11.0DR3 In Situation 2.1 let x0 : Spec(k) → X be a morphism, where kis a finite type field over S. By Lemma 2.3 and Formal Deformation Theory, The-orem 26.4 we know that FX ,k,x0 has a presentation by a smooth prorepresentablegroupoid in functors on CΛ. Unwinding the definitions, this means we can choose

(1) a Noetherian complete local Λ-algebra A with residue field k and a versalformal object ξ of FX ,k,x0 over A,

(2) a Noetherian complete local Λ-algebra B with residue field k and an iso-morphism

B|CΛ −→ A|CΛ ×ξ,FX ,k,x0 ,ξA|CΛ

The projections correspond to formally smooth maps t : A → B and s : A → B(because ξ is versal). There is a map c : B → B⊗s,A,tB which turns (A,B, s, t, c)into a cogroupoid in the category of Noetherian complete local Λ-algebras withresidue field k (on prorepresentable functors this map is constructed in FormalDeformation Theory, Lemma 25.2). Finally, the cited theorem tells us that ξ inducesan equivalence

[A|CΛ/B|CΛ ] −→ FX ,k,x0

of groupoids cofibred over CΛ. In fact, we also get an equivalence

[A/B] −→ FX ,k,x0

of groupoids cofibred over the completed category CΛ (see discussion in FormalDeformation Theory, Section 22 as to why this works). Of course A is a versal ringto X at x0.

3. Multiplicities of components of algebraic stacks

0DR4 If X is a locally Noetherian scheme, then we may write X (thought of simply asa topological space) as a union of irreducible components, say X =

⋃Ti. Each

irreducible component is the closure of a unique generic point ξi, and the local ringOX,ξi is a local Artin ring. We may define the multiplicity of X along Ti or themultiplicity of Ti in X by

mTi,X = lengthOX,ξiOX,ξiIn other words, it is the length of the local Artinian ring. Please compare withChow Homology, Section 9.Our goal here is to generalise this definition to locally Noetherian algebraic stacks.If X is a stack, then its topological space |X | (see Properties of Stacks, Definition4.8) is locally Noetherian (Morphisms of Stacks, Lemma 8.3). The irreduciblecomponents of |X | are sometimes referred to as the irreducible components of X . IfX is quasi-separated, then |X | is sober (Morphisms of Stacks, Lemma 30.3), but it

THE GEOMETRY OF ALGEBRAIC STACKS 6

need not be in the non-quasi-separated case. Consider for example the non-quasi-separated algebraic space X = A1

C/Z. Furthermore, there is no structure sheaf on|X | whose stalks can be used to define multiplicities.

Lemma 3.1.0DR5 Let f : U → X be a smooth morphism from a scheme to a locallyNoetherian algebraic stack. The closure of the image of any irreducible componentof |U | is an irreducible component of |X |. If U → X is surjective, then all irreduciblecomponents of |X | are obtained in this way.

Proof. The map |U | → |X | is continuous and open by Properties of Stacks, Lemma4.7. Let T ⊂ |U | be an irreducible component. Since U is locally Noetherian, wecan find a nonempty affine open W ⊂ U contained in T . Then f(T ) ⊂ |X | isirreducible and contains the nonempty open subset f(W ). Thus the closure off(T ) is irreducible and contains a nonempty open. It follows that this closure is anirreducible component.

Assume U → X is surjective and let Z ⊂ |X | be an irreducible component. Choosea Noetherian open subset V of |X | meeting Z. After removing the other irreduciblecomponents from V we may assume that V ⊂ Z. Take an irreducible componentof the nonempty open f−1(V ) ⊂ |U | and let T ⊂ |U | be its closure. This is anirreducible component of |U | and the closure of f(T ) must agree with Z by ourchoice of T .

The preceding lemma applies in particular in the case of smooth morphisms be-tween locally Noetherian schemes. This particular case is implicitly invoked in thestatement of the following lemma.

Lemma 3.2.0DR6 Let U → X be a smooth morphism of locally Noetherian schemes.Let T ′ is an irreducible component of U . Let T be the irreducible component of Xobtained as the closure of the image of T ′. Then mT ′,U = mT,X .

Proof. Write ξ′ for the generic point of T ′, and ξ for the generic point of T . LetA = OX,ξ and B = OU,ξ′ . We need to show that lengthAA = lengthBB. SinceA → B is a flat local homomorphism of rings (since smooth morphisms are flat),we have

lengthA(A)lengthB(B/mAB) = lengthB(B)by Algebra, Lemma 51.13. Thus it suffices to show mAB = mB , or equivalently, thatB/mAB is reduced. Since U → X is smooth, so is its base change Uξ → Specκ(ξ).As Uξ is a smooth scheme over a field, it is reduced, and thus so its local ring atany point (Varieties, Lemma 25.4). In particular,

B/mAB = OU,ξ′/mX,ξOU,ξ′ = OUξ,ξ′

is reduced, as required.

Using this result, we may show that there exists a good notion of multiplicity bylooking smooth locally.

Lemma 3.3.0DR7 Let U1 → X and U2 → X be two smooth morphisms from schemesto a locally Noetherian algebraic stack X . Let T ′1 and T ′2 be irreducible componentsof |U1| and |U2| respectively. Assume the closures of the images of T ′1 and T ′2 arethe same irreducible component T of |X |. Then mT ′

1,U1 = mT ′2,U2 .

THE GEOMETRY OF ALGEBRAIC STACKS 7

Proof. Let V1 and V2 be dense subsets of T ′1 and T ′2, respectively, that are openin U1 and U2 respectively (see proof of Lemma 3.1). The images of |V1| and |V2|in |X | are non-empty open subsets of the irreducible subset T , and therefore havenon-empty intersection. By Properties of Stacks, Lemma 4.3, the map |V1×X V2| →|V1| ×|X | |V2| is surjective. Consequently V1 ×X V2 is a non-empty algebraic space;we may therefore choose an étale surjection V → V1 ×X V2 whose source is a (non-empty) scheme. If we let T ′ be any irreducible component of V , then Lemma 3.1shows that the closure of the image of T ′ in U1 (respectively U2) is equal to T ′1(respectively T ′2).Applying Lemma 3.2 twice we find that

mT ′1,U1 = mT ′,V = mT ′

2,U2 ,

as required.

At this point we have done enough work to show the following definition makessense.

Definition 3.4.0DR8 Let X be a locally Noetherian algebraic stack. Let T ⊂ |X | bean irreducible component. The multiplicity of T in X is defined as mT,X = mT ′,U

where f : U → X is a smooth morphism from a scheme and T ′ ⊂ |U | is anirreducible component with f(T ′) ⊂ T .

This is independent of the choice of f : U → X and the choice of the irreduciblecomponent T ′ mapping to T by Lemmas 3.1 and 3.3.As a closing remark, we note that it is sometimes convenient to think of an irre-ducible component of X as a closed substack. To this end, if T is an irreduciblecomponent of X , i.e., an irreducible component of |X |, then we endow T with itsinduced reduced substack structure, see Properties of Stacks, Definition 10.4.

4. Formal branches and multiplicities

0DR9 It will be convenient to have a comparison between the notion of multiplicity of anirreducible component given by Definition 3.4 and the related notion of multiplicitiesof irreducible components of (the spectra of) versal rings of X at finite type points.In Situation 2.1 let x0 : Spec(k)→ X be a morphism, where k is a finite type fieldover S. Let A, A′ be versal rings to X at x0. After possibly swapping A and A′, weknow there is a formally smooth1 map ϕ : A → A′ compatible with versal formalobjects, see Lemma 2.4 and Remark 2.9. Moreover, ϕ is well defined up to formalhomotopy, see Formal Deformation Theory, Lemma 28.3. In particular, we findthat ϕ(p)A′ is a well defined ideal of A′ by Formal Deformation Theory, Lemma28.4. Since A → A′ is formally smooth, in fact ϕ(p)A′ is a minimal prime of A′and every minimal prime of A′ is of this form for a unique minimal prime p ⊂ A(all of this is easy to prove by writing A′ as a power series ring over A). Therefore,recalling that minimal primes correspond to irreducible components, the followingdefinition makes sense.

Definition 4.1.0DRA Let X be an algebraic stack locally of finite type over a locallyNoetherian scheme S. Let x0 : Spec(k) → X is a morphism where k is a field offinite type over S. The formal branches of X through x0 is the set of irreducible

1In the sense that A′ becomes isomorphic to a power series ring over A.

THE GEOMETRY OF ALGEBRAIC STACKS 8

components of Spec(A) for any choice of versal ring to X at x0 identified for differentchoices of A by the procedure described above.Suppose in the situation of Definition 4.1 we are given a finite extension l/k. Setxl,0 : Spec(l) → X equal to the composition of Spec(l) → Spec(k) with x0. LetA→ A′ be as in Lemma 2.5. Since A→ A′ is faithfully flat, the morphism

Spec(A′)→ Spec(A)sends (generic points of) irreducible components to (generic points of) irreduciblecomponents. This will be a surjective map, but in general this map will not be abijection. In other words, we obtain a surjective map

formal branches of X through xl,0 −→ formal branches of X through x0

It turns out that if l/k is purely inseparable, then the map is injective as well (we’lladd a precise statement and proof here if we ever need this).Lemma 4.2.0DRB In the situation of Definition 4.1 there is a canonical surjection fromthe set of formal branches of X through x0 to the set of irreducible components of|X | containing x0 in |X |.Proof. Let A be as in Definition 4.1 and let Spec(A) → X be as in Remark 2.9.We claim that the generic point of an irreducible component of Spec(A) maps toa generic point of an irreducible component of |X |. Choose a scheme U and asurjective smooth morphism U → X . Consider the diagram

Spec(A)×X U

p

q// U

f

Spec(A) j // X

By Lemma 2.10 we see that j is flat. Hence q is flat. On the other hand, f issurjective smooth hence p is surjective smooth. This implies that any generic pointη ∈ Spec(A) of an irreducible component is the image of a codimension 0 point η′ ofthe algebraic space Spec(A)×X U (see Properties of Spaces, Section 11 for notationand use going down on étale local rings). Since q is flat, q(η′) is a codimension 0point of U (same argument). Since U is a scheme, q(η′) is the generic point of anirreducible component of U . Thus the closure of the image of q(η′) in |X | is anirreducible component by Lemma 3.1 as claimed.Clearly the claim provides a mechanism for defining the desired map. To see thatit is surjective, we choose u0 ∈ U mapping to x0 in |X |. Choose an affine openU ′ ⊂ U neighbourhood of u0. After shrinking U ′ we may assume every irreduciblecomponent of U ′ passes through u0. Then we may replace X by the open substackcorresponding to the image of |U ′| → |X |. Thus we may assume U is affine has apoint u0 mapping to x0 ∈ |X | and every irreducible component of U passes throughu0. By Properties of Stacks, Lemma 4.3 there is a point t ∈ | Spec(A) ×X U |mapping to the closed point of Spec(A) and to u0. Using going down for the flatlocal ring homomorphisms

A −→ OSpec(A)×XU,t←− OU,u0

we see that every minimal prime of OU,u0 is the image of a minimal prime of thelocal ring in the middle and such a minimal prime maps to a minimal prime of A.This proves the surjectivity. Some details omitted.

THE GEOMETRY OF ALGEBRAIC STACKS 9

Let A be a Noetherian complete local ring. Then the irreducible components ofSpec(A) have multiplicities, see introduction to Section 3. If A′ = A[[t1, . . . , tr]],then the morphism Spec(A′)→ Spec(A) induces a bijection on irreducible compo-nents preserving multiplicities (we omit the easy proof). This and the discussionpreceding Definition 4.1 mean that the following definition makes sense.

Definition 4.3.0DRC Let X be an algebraic stack locally of finite type over a locallyNoetherian scheme S. Let x0 : Spec(k) → X is a morphism where k is a fieldof finite type over S. The multiplicity of a formal branch of X through x0 is themultiplicity of the corresponding irreducible component of Spec(A) for any choiceof versal ring to X at x0 (see discussion above).

Lemma 4.4.0DRD Let X be an algebraic stack locally of finite type over a locallyNoetherian scheme S. Let x0 : Spec(k) → X is a morphism where k is a field offinite type over S with image s ∈ S. If OS,s is a G-ring, then the map of Lemma4.2 preserves multiplicities.

Proof. By Lemma 2.8 we may assume there is a smooth morphism U → X whereU is a scheme and a k-valued point u0 of U such that O∧U,u0

is a versal ring to Xat x0. By construction of our map in the proof of Lemma 4.2 (which simplifiesgreatly because A = O∧U,u0

) we find that it suffices to show: the multiplicity of anirreducible component of U passing through u0 is the same as the multiplicity ofany irreducible component of Spec(O∧U,u0

) mapping into it.

Translated into commutative algebra we find the following: Let C = OU,u0 . Thisis essentially of finite type over OS,s and hence is a G-ring (More on Algebra,Proposition 49.10). Then A = C∧. Therefore C → A is a regular ring map. Letq ⊂ C be a minimal prime and let p ⊂ A be a minimal prime lying over q. Then

R = Cp −→ Ap = R′

is a regular ring map of Artinian local rings. For such a ring map it is always thecase that

lengthRR = lengthR′R′

This is what we have to show because the left hand side is the multiplicity of ourcomponent on U and the right hand side is the multiplicity of our component onSpec(A). To see the equality, first we use that

lengthR(R)lengthR′(R′/mRR′) = lengthR′(R′)

by Algebra, Lemma 51.13. Thus it suffices to show mRR′ = mR′ , which is a

consequence of being a regular homomorphism of zero dimensional local rings.

5. Dimension theory of algebraic stacks

0DRE The main results on the dimension theory of algebraic stacks in the literature thatwe are aware of are those of [Oss15], which makes a study of the notions of codimen-sion and relative dimension. We make a more detailed examination of the notionof the dimension of an algebraic stack at a point, and prove various results relatingthe dimension of the fibres of a morphism at a point in the source to the dimensionof its source and target. We also prove a result (Lemma 6.4 below) which allow us(under suitable hypotheses) to compute the dimension of an algebraic stack at apoint in terms of a versal ring.

THE GEOMETRY OF ALGEBRAIC STACKS 10

While we haven’t always tried to optimise our results, we have largely tried toavoid making unnecessary hypotheses. However, in some of our results, in whichwe compare certain properties of an algebraic stack to the properties of a versal ringto this stack at a point, we have restricted our attention to the case of algebraicstacks that are locally finitely presented over a locally Noetherian scheme base,all of whose local rings are G-rings. This gives us the convenience of having Artinapproximation available to compare the geometry of the versal ring to the geometryof the stack itself. However, this restrictive hypothesis may not be necessary forthe truth of all of the various statements that we prove. Since it is satisfied in theapplications that we have in mind, though, we have been content to make it whenit helps.

If X is a scheme, then we define the dimension dim(X) of X to be the Krulldimension of the topological space underlying X, while if x is a point of X, then wedefine the dimension dimx(X) of X at x to be the minimum of the dimensions ofthe open subsets U of X containing x, see Properties, Definition 10.1. One has therelation dim(X) = supx∈X dimx(X), see Properties, Lemma 10.2. If X is locallyNoetherian, then dimx(X) coincides with the supremum of the dimensions at x ofthe irreducible components of X passing through x.

If X is an algebraic space and x ∈ |X|, then we define dimxX = dimu U, where U isany scheme admitting an étale surjection U → X, and u ∈ U is any point lying overx, see Properties of Spaces, Definition 9.1. We set dim(X) = supx∈|X| dimx(X),see Properties of Spaces, Definition 9.2.

Remark 5.1.0DRF In general, the dimension of the algebraic space X at a point x maynot coincide with the dimension of the underlying topological space |X| at x. E.g.if k is a field of characteristic zero and X = A1

k/Z, then X has dimension 1 (thedimension of A1

k) at each of its points, while |X| has the indiscrete topology, andhence is of Krull dimension zero. On the other hand, in Algebraic Spaces, Example14.9 there is given an example of an algebraic space which is of dimension 0 at eachof its points, while |X| is irreducible of Krull dimension 1, and admits a genericpoint (so that the dimension of |X| at any of its points is 1); see also the discussionof this example in Properties of Spaces, Section 9.

On the other hand, if X is a decent algebraic space, in the sense of Decent Spaces,Definition 6.1 (in particular, if X is quasi-separated; see Decent Spaces, Section 6)then in fact the dimension of X at x does coincide with the dimension of |X| at x;see Decent Spaces, Lemma 12.5.

In order to define the dimension of an algebraic stack, it will be useful to first havethe notion of the relative dimension, at a point in the source, of a morphism whosesource is an algebraic space, and whose target is an algebraic stack. The definitionis slightly involved, just because (unlike in the case of schemes) the points of analgebraic stack, or an algebraic space, are not describable as morphisms from thespectrum of a field, but only as equivalence classes of such.

Definition 5.2.0DRG If f : T → X is a locally of finite type morphism from an algebraicspace to an algebraic stack, and if t ∈ |T | is a point with image x ∈ |X |, then wedefine the relative dimension of f at t, denoted dimt(Tx), as follows: choose amorphism Spec k → X , with source the spectrum of a field, which represents x,

THE GEOMETRY OF ALGEBRAIC STACKS 11

and choose a point t′ ∈ |T ×X Spec k| mapping to t under the projection to |T |(such a point t′ exists, by Properties of Stacks, Lemma 4.3); then

dimt(Tx) = dimt′(T ×X Spec k).

Note that since T is an algebraic space and X is an algebraic stack, the fibre productT ×X Spec k is an algebraic space, and so the quantity on the right hand side ofthis proposed definition is in fact defined (see discussion above).

Remark 5.3.0DRH (1) One easily verifies (for example, by using the invariance ofthe relative dimension of locally of finite type morphisms of schemes under base-change; see for example Morphisms, Lemma 28.3) that dimt(Tx) is well-defined,independently of the choices used to compute it.(2) In the case that X is also an algebraic space, it is straightforward to confirm thatthis definition agrees with the definition of relative dimension given in Morphismsof Spaces, Definition 33.1.

We next recall the following lemma, on which our study of the dimension of a locallyNoetherian algebraic stack is founded.

Lemma 5.4.0DRI If f : U → X is a smooth morphism of locally Noetherian algebraicspaces, and if u ∈ |U | with image x ∈ |X|, then

dimu(U) = dimx(X) + dimu(Ux)where dimu(Ux) is defined via Definition 5.2.

Proof. See Morphisms of Spaces, Lemma 37.10 noting that the definition of dimu(Ux)used here coincides with the definition used there, by Remark 5.3 (2).

Lemma 5.5.0DRJ If X is a locally Noetherian algebraic stack and x ∈ |X |. Let U → Xbe a smooth morphism from an algebraic space to X , let u be any point of |U |mapping to x. Then we have

dimx(X ) = dimu(U)− dimu(Ux)where the relative dimension dimu(Ux) is defined by Definition 5.2 and the dimen-sion of X at x is as in Properties of Stacks, Definition 12.2.

Proof. Lemma 5.4 can be used to verify that the right hand side dimu(U) +dimu(Ux) is independent of the choice of the smooth morphism U → X and u ∈ |U |.We omit the details. In particular, we may assume U is a scheme. In this case wecan compute dimu(Ux) by choosing the representative of x to be the compositeSpecκ(u) → U → X , where the first morphism is the canonical one with imageu ∈ U . Then, if we write R = U ×X U , and let e : U → R denote the di-agonal morphism, the invariance of relative dimension under base-change showsthat dimu(Ux) = dime(u)(Ru). Thus we see that the right hand side is equal todimu(U)− dime(u)(Ru) = dimx(X ) as desired.

Remark 5.6.0DRK For Deligne–Mumford stacks which are suitably decent (e.g. quasi-separated), it will again be the case that dimx(X ) coincides with the topologicallydefined quantity dimx |X |. However, for more general Artin stacks, this will typ-ically not be the case. For example, if X = [A1/Gm] (over some field, with thequotient being taken with respect to the usual multiplication action of Gm on A1),then |X | has two points, one the specialisation of the other (corresponding to the

THE GEOMETRY OF ALGEBRAIC STACKS 12

two orbits of Gm on A1), and hence is of dimension 1 as a topological space; butdimx(X ) = 0 for both points x ∈ |X |. (An even more extreme example is given bythe classifying space [Spec k/Gm], whose dimension at its unique point is equal to−1.)

We can now extend Definition 5.2 to the context of (locally finite type) morphismsbetween (locally Noetherian) algebraic stacks.

Definition 5.7.0DRL If f : T → X is a locally of finite type morphism between locallyNoetherian algebraic stacks, and if t ∈ |T | is a point with image x ∈ |X |, thenwe define the relative dimension of f at t, denoted dimt(Tx), as follows: choose amorphism Spec k → X , with source the spectrum of a field, which represents x, andchoose a point t′ ∈ |T ×X Spec k| mapping to t under the projection to |T | (such apoint t′ exists, by Properties of Stacks, Lemma 4.3; then

dimt(Tx) = dimt′(T ×X Spec k).

Note that since T is an algebraic stack and X is an algebraic stack, the fibre productT ×X Spec k is an algebraic stack, which is locally Noetherian by Morphisms ofStacks, Lemma 17.5. Thus the quantity on the right side of this proposed definitionis defined by Properties of Stacks, Definition 12.2.

Remark 5.8.0DRM Standard manipulations show that dimt(Tx) is well-defined, inde-pendently of the choices made to compute it.

We now establish some basic properties of relative dimension, which are obviousgeneralisations of the corresponding statements in the case of morphisms of schemes.

Lemma 5.9.0DRN Suppose given a Cartesian square of morphisms of locally Noetherianstacks

T ′

// T

X ′ // X

in which the vertical morphisms are locally of finite type. If t′ ∈ |T ′|, with imagest, x′, and x in |T |, |X ′|, and |X | respectively, then dimt′(T ′x′) = dimt(Tx).

Proof. Both sides can (by definition) be computed as the dimension of the samefibre product.

Lemma 5.10.0DRP If f : U → X is a smooth morphism of locally Noetherian algebraicstacks, and if u ∈ |U| with image x ∈ |X |, then

dimu(U) = dimx(X ) + dimu(Ux).

Proof. Choose a smooth surjective morphism V → U whose source is a scheme,and let v ∈ |V | be a point mapping to u. Then the composite V → U → X isalso smooth, and by Lemma 5.4 we have dimx(X ) = dimv(V ) − dimv(Vx), whiledimu(U) = dimv(V )− dimv(Vu). Thus

dimu(U)− dimx(X ) = dimv(Vx)− dimv(Vu).

Choose a representative Spec k → X of x and choose a point v′ ∈ |V ×X Spec k| lyingover v, with image u′ in |U ×X Spec k|; then by definition dimu(Ux) = dimu′(U ×XSpec k), and dimv(Vx) = dimv′(V ×X Spec k).

THE GEOMETRY OF ALGEBRAIC STACKS 13

Now V ×X Spec k → U ×X Spec k is a smooth surjective morphism (being the base-change of such a morphism) whose source is an algebraic space (since V and Spec kare schemes, and X is an algebraic stack). Thus, again by definition, we have

dimu′(U ×X Spec k) = dimv′(V ×X Spec k)− dimv′(V ×X Spec k)u′)= dimv(Vx)− dimv′((V ×X Spec k)u′).

Now V ×X Spec k ∼= V ×U (U×X Spec k), and so Lemma 5.9 shows that dimv′((V ×XSpec k)u′) = dimv(Vu). Putting everything together, we find that

dimu(U)− dimx(X ) = dimu(Ux),as required.

Lemma 5.11.0DRQ Let f : T → X be a locally of finite type morphism of algebraicstacks.

(1) The function t 7→ dimt(Tf(t)) is upper semi-continuous on |T |.(2) If f is smooth, then the function t 7→ dimt(Tf(t)) is locally constant on |T |.

Proof. Suppose to begin with that T is a scheme T , let U → X be a smoothsurjective morphism whose source is a scheme, and let T ′ = T×XU . Let f ′ : T ′ → Ube the pull-back of f over U , and let g : T ′ → T be the projection.Lemma 5.9 shows that dimt′(T ′f ′(t′)) = dimg(t′)(Tf(g(t′))), for t′ ∈ T ′, while, since gis smooth and surjective (being the base-change of a smooth surjective morphism)the map induced by g on underlying topological spaces is continuous and open (byProperties of Spaces, Lemma 4.6), and surjective. Thus it suffices to note that part(1) for the morphism f ′ follows from Morphisms of Spaces, Lemma 34.4, and part(2) from either of Morphisms, Lemma 29.4 or Morphisms, Lemma 33.12 (each ofwhich gives the result for schemes, from which the analogous results for algebraicspaces can be deduced exactly as in Morphisms of Spaces, Lemma 34.4.Now return to the general case, and choose a smooth surjective morphism h : V →T whose source is a scheme. If v ∈ V , then, essentially by definition, we have

dimh(v)(Tf(h(v))) = dimv(Vf(h(v)))− dimv(Vh(v)).Since V is a scheme, we have proved that the first of the terms on the right handside of this equality is upper semi-continuous (and even locally constant if f issmooth), while the second term is in fact locally constant. Thus their difference isupper semi-continuous (and locally constant if f is smooth), and hence the func-tion dimh(v)(Tf(h(v))) is upper semi-continuous on |V | (and locally constant if fis smooth). Since the morphism |V | → |T | is open and surjective, the lemmafollows.

Before continuing with our development, we prove two lemmas related to the di-mension theory of schemes.To put the first lemma in context, we note that if X is a finite dimensional scheme,then since dimX is defined to equal the supremum of the dimensions dimxX, thereexists a point x ∈ X such that dimxX = dimX. The following lemma shows thatwe may furthermore take the point x to be of finite type.

Lemma 5.12.0DRR If X is a finite dimensional scheme, then there exists a closed (andhence finite type) point x ∈ X such that dimxX = dimX.

THE GEOMETRY OF ALGEBRAIC STACKS 14

Proof. Let d = dimX, and choose a maximal strictly decreasing chain of irre-ducible closed subsets of X, say

(5.12.1)0DRS Z0 ⊃ Z1 ⊃ . . . ⊃ Zd.

The subset Zd is a minimal irreducible closed subset of X, and thus any point ofZd is a generic point of Zd. Since the underlying topological space of the schemeX is sober, we conclude that Zd is a singleton, consisting of a single closed pointx ∈ X. If U is any neighbourhood of x, then the chain

U ∩ Z0 ⊃ U ∩ Z1 ⊃ . . . ⊃ U ∩ Zd = Zd = x

is then a strictly descending chain of irreducible closed subsets of U , showing thatdimU ≥ d. Thus we find that dimxX ≥ d. The other inequality being obvious,the lemma is proved.

The next lemma shows that dimxX is a constant function on an irreducible schemesatisfying some mild additional hypotheses.

Lemma 5.13.0DRT If X is an irreducible, Jacobson, catenary, and locally Noetherianscheme of finite dimension, then dimU = dimX for every non-empty open subsetU of X. Equivalently, dimxX is a constant function on X.

Proof. The equivalence of the two claims follows directly from the definitions.Suppose, then, that U ⊂ X is a non-empty open subset. Certainly dimU ≤ dimX,and we have to show that dimU ≥ dimX.Write d = dimX, and choose a maximalstrictly decreasing chain of irreducible closed subsets of X, say

X = Z0 ⊃ Z1 ⊃ . . . ⊃ Zd.

Since X is Jacobson, the minimal irreducible closed subset Zd is equal to x forsome closed point x.

If x ∈ U, thenU = U ∩ Z0 ⊃ U ∩ Z1 ⊃ . . . ⊃ U ∩ Zd = x

is a strictly decreasing chain of irreducible closed subsets of U , and so we concludethat dimU ≥ d, as required. Thus we may suppose that x 6∈ U.

Consider the flat morphism SpecOX,x → X. The non-empty (and hence dense)open subset U of X pulls back to an open subset V ⊂ SpecOX,x. Replacing U bya non-empty quasi-compact, and hence Noetherian, open subset, we may assumethat the inclusion U → X is a quasi-compact morphism. Since the formationof scheme-theoretic images of quasi-compact morphisms commutes with flat base-change Morphisms, Lemma 25.16 we see that V is dense in SpecOX,x, and so inparticular non-empty, and of course x 6∈ V. (Here we use x also to denote theclosed point of SpecOX,x, since its image is equal to the given point x ∈ X.)Now SpecOX,x \ x is Jacobson Properties, Lemma 6.4 and hence V contains aclosed point z of SpecOX,x \ x. The closure in X of the image of z is then anirreducible closed subset Z of X containing x, whose intersection with U is non-empty, and for which there is no irreducible closed subset properly contained in Zand properly containing x (because pull-back to SpecOX,x induces a bijectionbetween irreducible closed subsets of X containing x and irreducible closed subsetsof SpecOX,x). Since U ∩ Z is a non-empty closed subset of U , it contains a pointu that is closed in X (since X is Jacobson), and since U ∩ Z is a non-empty (and

THE GEOMETRY OF ALGEBRAIC STACKS 15

hence dense) open subset of the irreducible set Z (which contains a point not lyingin U , namely x), the inclusion u ⊂ U ∩ Z is proper.

As X is catenary, the chain

X = Z0 ⊃ Z ⊃ x = Zd

can be refined to a chain of length d+ 1, which must then be of the form

X = Z0 ⊃W1 ⊃ . . . ⊃Wd−1 = Z ⊃ x = Zd.

Since U ∩ Z is non-empty, we then find that

U = U ∩ Z0 ⊃ U ∩W1 ⊃ . . . ⊃ U ∩Wd−1 = U ∩ Z ⊃ u

is a strictly decreasing chain of irreducible closed subsets of U of length d + 1,showing that dimU ≥ d, as required.

We will prove a stack-theoretic analogue of Lemma 5.13 in Lemma 5.17 below, butbefore doing so, we have to introduce an additional definition, necessitated by thefact that the notion of a scheme being catenary is not an étale local one (see theexample of Algebra, Remark 162.8 which makes it difficult to define what it meansfor an algebraic space or algebraic stack to be catenary (see the discussion of [Oss15,page 3]). For certain aspects of dimension theory, the following definition seems toprovide a good substitute for the missing notion of a catenary algebraic stack.

Definition 5.14.0DRU We say that a locally Noetherian algebraic stack X is pseudo-catenary if there exists a smooth and surjective morphism U → X whose source isa universally catenary scheme.

Example 5.15.0DRV If X is locally of finite type over a universally catenary locallyNoetherian scheme S, and U → X is a smooth surjective morphism whose sourceis a scheme, then the composite U → X → S is locally of finite type, and so U isuniversally catenary Morphisms, Lemma 17.2. Thus X is pseudo-catenary.

The following lemma shows that the property of being pseudo-catenary passesthrough finite-type morphisms.

Lemma 5.16.0DRW If X is a pseudo-catenary locally Noetherian algebraic stack, andif Y → X is a locally of finite type morphism, then there exists a smooth surjectivemorphism V → Y whose source is a universally catenary scheme; thus Y is againpseudo-catenary.

Proof. By assumption we may find a smooth surjective morphism U → X whosesource is a universally catenary scheme. The base-change U ×X Y is then an al-gebraic stack; let V → U ×X Y be a smooth surjective morphism whose sourceis a scheme. The composite V → U ×X Y → Y is then smooth and surjec-tive (being a composite of smooth and surjective morphisms), while the morphismV → U ×X Y → U is locally of finite type (being a composite of morphisms thatare locally finite type). Since U is universally catenary, we see that V is universallycatenary (by Morphisms, Lemma 17.2), as claimed.

We now study the behaviour of the function dimx(X ) on |X | (for some locallyNoetherian stack X ) with respect to the irreducible components of |X |, as well asvarious related topics.

THE GEOMETRY OF ALGEBRAIC STACKS 16

Lemma 5.17.0DRX If X is a Jacobson, pseudo-catenary, and locally Noetherian al-gebraic stack for which |X | is irreducible, then dimx(X ) is a constant function on|X |.

Proof. It suffices to show that dimx(X ) is locally constant on |X |, since it willthen necessarily be constant (as |X | is connected, being irreducible). Since X ispseudo-catenary, we may find a smooth surjective morphism U → X with U beinga universally catenary scheme. If Ui is an cover of U by quasi-compact opensubschemes, we may replace U by

∐Ui,, and it suffices to show that the function

u 7→ dimf(u)(X ) is locally constant on Ui. Since we check this for one Ui at atime, we now drop the subscript, and write simply U rather than Ui. Since U isquasi-compact, it is the union of a finite number of irreducible components, sayT1 ∪ . . . ∪ Tn. Note that each Ti is Jacobson, catenary, and locally Noetherian,being a closed subscheme of the Jacobson, catenary, and locally Noetherian schemeU .

By Lemma 5.4, we have dimf(u)(X ) = dimu(U) − dimu(Uf(u)). Lemma 5.11 (2)shows that the second term in the right hand expression is locally constant on U ,as f is smooth, and hence we must show that dimu(U) is locally constant on U .Since dimu(U) is the maximum of the dimensions dimu Ti, as Ti ranges over thecomponents of U containing u, it suffices to show that if a point u lies on twodistinct components, say Ti and Tj (with i 6= j), then dimu Ti = dimu Tj , and thento note that t 7→ dimt T is a constant function on an irreducible Jacobson, catenary,and locally Noetherian scheme T (as follows from Lemma 5.13).

Let V = Ti \ (⋃i′ 6=i Ti′) and W = Tj \ (

⋃i′ 6=j Ti′). Then each of V and W is

a non-empty open subset of U , and so each has non-empty open image in |X |.As |X | is irreducible, these two non-empty open subsets of |X | have a non-emptyintersection. Let x be a point lying in this intersection, and let v ∈ V and w ∈ Wbe points mapping to x. We then find that

dimTi = dimV = dimv(U) = dimx(X ) + dimv(Ux)

and similarly that

dimTj = dimW = dimw(U) = dimx(X ) + dimw(Ux).

Since u 7→ dimu(Uf(u)) is locally constant on U , and since Ti ∪ Tj is connected(being the union of two irreducible, hence connected, sets that have non-empty in-tersection), we see that dimv(Ux) = dimw(Ux), and hence, comparing the precedingtwo equations, that dimTi = dimTj , as required.

Lemma 5.18.0DRY If Z → X is a closed immersion of locally Noetherian schemes,and if z ∈ |Z| has image x ∈ |X |, then dimz(Z) ≤ dimx(X ).

Proof. Choose a smooth surjective morphism U → X whose source is a scheme;the base-changed morphism V = U ×X Z → Z is then also smooth and surjective,and the projection V → U is a closed immersion. If v ∈ |V | maps to z ∈ |Z|, andif we let u denote the image of v in |U |, then clearly dimv(V ) ≤ dimu(U), whiledimv(Vz) = dimu(Ux), by Lemma 5.9. Thus

dimz(Z) = dimv(V )− dimv(Vz) ≤ dimu(U)− dimu(Ux) = dimx(X ),

as claimed.

THE GEOMETRY OF ALGEBRAIC STACKS 17

Lemma 5.19.0DRZ If X is a locally Noetherian algebraic stack, and if x ∈ |X |, thendimx(X ) = supT dimx(T ), where T runs over all the irreducible components of|X | passing through x (endowed with their induced reduced structure).

Proof. Lemma 5.18 shows that dimx(T ) ≤ dimx(X ) for each irreducible compo-nent T passing through the point x. Thus to prove the lemma, it suffices to showthat

(5.19.1)0DS0 dimx(X ) ≤ supTdimx(T ).

Let U → X be a smooth cover by a scheme. If T is an irreducible componentof U then we let T denote the closure of its image in X , which is an irreduciblecomponent of X . Let u ∈ U be a point mapping to x. Then we have dimx(X ) =dimu U − dimu Ux = supT dimu T − dimu Ux, where the supremum is over theirreducible components of U passing through u. Choose a component T for whichthe supremum is achieved, and note that dimx(T ) = dimu T−dimu Tx. The desiredinequality (5.19.1) now follows from the evident inequality dimu Tx ≤ dimu Ux.(Note that if Spec k → X is a representative of x, then T ×X Spec k is a closedsubspace of U ×X Spec k.)

Lemma 5.20.0DS1 If X is a locally Noetherian algebraic stack, and if x ∈ |X |, thenfor any open substack V of X containing x, there is a finite type point x0 ∈ |V| suchthat dimx0(X ) = dimx(V).

Proof. Choose a smooth surjective morphism f : U → X whose source is a scheme,and consider the function u 7→ dimf(u)(X ); since the morphism |U | → |X | inducedby f is open (as f is smooth) as well as surjective (by assumption), and takes finitetype points to finite type points (by the very definition of the finite type points of|X |), it suffices to show that for any u ∈ U , and any open neighbourhood of u, thereis a finite type point u0 in this neighbourhood such that dimf(u0)(X ) = dimf(u)(X ).Since, with this reformulation of the problem, the surjectivity of f is no longerrequired, we may replace U by the open neighbourhood of the point u in question,and thus reduce to the problem of showing that for each u ∈ U , there is a finite typepoint u0 ∈ U such that dimf(u0)(X ) = dimf(u)(X ). By Lemma 5.4 dimf(u)(X ) =dimu(U) − dimu(Uf(u)), while dimf(u0)(X ) = dimu0(U) − dimu0(Uf(u0)). Since fis smooth, the expression dimu0(Uf(u0)) is locally constant as u0 varies over U (byLemma 5.11 (2)), and so shrinking U further around u if necessary, we may assumeit is constant. Thus the problem becomes to show that we may find a finite typepoint u0 ∈ U for which dimu0(U) = dimu(U). Since by definition dimu U is theminimum of the dimensions dimV , as V ranges over the open neighbourhoods Vof u in U , we may shrink U down further around u so that dimu U = dimU . Theexistence of desired point u0 then follows from Lemma 5.12.

Lemma 5.21.0DS2 Let T → X be a locally of finite type monomorphism of algebraicstacks, with X (and thus also T ) being Jacobson, pseudo-catenary, and locally Noe-therian. Suppose further that T is irreducible of some (finite) dimension d, andthat X is reduced and of dimension less than or equal to d. Then there is a non-empty open substack V of T such that the induced monomorphism V → X is anopen immersion which identifies V with an open subset of an irreducible componentof X .

THE GEOMETRY OF ALGEBRAIC STACKS 18

Proof. Choose a smooth surjective morphism f : U → X with source a scheme,necessarily reduced since X is, and write U ′ = T ×XU . The base-changed morphismU ′ → U is a monomorphism of algebraic spaces, locally of finite type, and thusrepresentable Morphisms of Spaces, Lemma 51.1 and 27.10; since U is a scheme,so is U ′. The projection f ′ : U ′ → T is again a smooth surjection. Let u′ ∈ U ′,with image u ∈ U . Lemma 5.9 shows that dimu′(U ′f(u′)) = dimu(Uf(u)), whiledimf ′(u′)(T ) = d ≥ dimf(u)(X ) by Lemma 5.17 and our assumptions on T and X .Thus we see that(5.21.1)

0DS3 dimu′(U ′) = dimu′(U ′f(u′))+dimf ′(u′)(T ) ≥ dimu(Uf(u))+dimf(u)(X ) = dimu(U).

Since U ′ → U is a monomorphism, locally of finite type, it is in particular unrami-fied, and so by the étale local structure of unramified morphisms Étale Morphisms,Lemma 17.3, we may find a commutative diagram

V ′ //

V

U ′ // U

in which the scheme V ′ is non-empty, the vertical arrows are étale, and the upperhorizontal arrow is a closed immersion. Replacing V by a quasi-compact open subsetwhose image has non-empty intersection with the image of U ′, and replacing V ′ bythe preimage of V , we may further assume that V (and thus V ′) is quasi-compact.Since V is also locally Noetherian, it is thus Noetherian, and so is the union offinitely many irreducible components.

Since étale morphisms preserve pointwise dimension Descent, Lemma 18.2 we de-duce from (5.21.1) that for any point v′ ∈ V ′, with image v ∈ V , we havedimv′(V ′) ≥ dimv(V ). In particular, the image of V ′ can’t be contained in theintersection of two distinct irreducible components of V , and so we may find atleast one irreducible open subset of V which has non-empty intersection with V ′;replacing V by this subset, we may assume that V is integral (being both reducedand irreducible). From the preceding inequality on dimensions, we conclude thatthe closed immersion V ′ → V is in fact an isomorphism. If we let W denote theimage of V ′ in U ′, thenW is a non-empty open subset of U ′ (as étale morphisms areopen), and the induced monomorphism W → U is étale (since it is so étale locallyon the source, i.e. after pulling back to V ′), and hence is an open immersion (beingan étale monomorphism). Thus, if we let V denote the image of W in T , then Vis a dense (equivalently, non-empty) open substack of T , whose image is dense inan irreducible component of X . Finally, we note that the morphism is V → X issmooth (since its composite with the smooth morphism W → V is smooth), andalso a monomorphism, and thus is an open immersion.

Lemma 5.22.0DS4 Let f : T → X be a locally of finite type morphism of Jacobson,pseudo-catenary, and locally Noetherian algebraic stacks, whose source is irreducibleand whose target is quasi-separated, and let Z → X denote the scheme-theoreticimage of T . Then for every finite type point t ∈ |T |, we have that dimt(Tf(t)) ≥dim T − dimZ, and there is a non-empty (equivalently, dense) open subset of |T |over which equality holds.

THE GEOMETRY OF ALGEBRAIC STACKS 19

Proof. Replacing X by Z, we may and do assume that f is scheme theoreticallydominant, and also that X is irreducible. By the upper semi-continuity of fibredimensions (Lemma 5.11 (1)), it suffices to prove that the equality dimt(Tf(t)) =dim T − dimZ holds for t lying in some non-empty open substack of T . For thisreason, in the argument we are always free to replace T by a non-empty opensubstack.

Let T ′ → T be a smooth surjective morphism whose source is a scheme, and let Tbe a non-empty quasi-compact open subset of T ′. Since Y is quasi-separated, wefind that T → Y is quasi-compact (by Morphisms of Stacks, Lemma 7.7, appliedto the morphisms T → Y → Spec Z). Thus, if we replace T by the image of T inT , then we may assume (appealing to Morphisms of Stacks, Lemma 7.6 that themorphism f : T → X is quasi-compact.

If we choose a smooth surjection U → X with U a scheme, then Lemma 3.1 ensuresthat we may find an irreducible open subset V of U such that V → X is smoothand scheme-theoretically dominant. Since scheme-theoretic dominance for quasi-compact morphisms is preserved by flat base-change, the base-change T ×X V →V of the scheme-theoretically dominant morphism f is again scheme-theoreticallydominant. We let Z denote a scheme admitting a smooth surjection onto this fibreproduct; then Z → T ×X V → V is again scheme-theoretically dominant. Thus wemay find an irreducible component C of Z which scheme-theoretically dominatesV . Since the composite Z → T ×X V → T is smooth, and since T is irreducible,Lemma 3.1 shows that any irreducible component of the source has dense imagein |T |. We now replace C by a non-empty open subset W which is disjoint fromevery other irreducible component of Z, and then replace T and X by the imagesof W and V (and apply Lemma 5.17 to see that this doesn’t change the dimensionof either T or X ). If we let W denote the image of the morphism W → T ×X V ,then W is open in T ×X V (since the morphism W → T ×X V is smooth), andis irreducible (being the image of an irreducible scheme). Thus we end up with acommutative diagram

W

!!

// W //

V

T // X

in which W and V are schemes, the vertical arrows are smooth and surjective,the diagonal arrows and the left-hand upper horizontal arroware smooth, and theinduced morphism W → T ×X V is an open immersion. Using this diagram,together with the definitions of the various dimensions involved in the statementof the lemma, we will reduce our verification of the lemma to the case of schemes,where it is known.

Fix w ∈ |W | with image w′ ∈ |W|, image t ∈ |T |, image v in |V |, and image x in|X |. Essentially by definition (using the fact that W is open in T ×X V , and thatthe fibre of a base-change is the base-change of the fibre), we obtain the equalities

dimv Vx = dimw′Wt

anddimt Tx = dimw′Wv.

THE GEOMETRY OF ALGEBRAIC STACKS 20

By Lemma 5.4 (the diagonal arrow and right-hand vertical arrow in our diagramrealise W and V as smooth covers by schemes of the stacks T and X ), we find that

dimt T = dimwW − dimwWt

anddimx X = dimv V − dimv Vx.

Combining the equalities, we find that

dimt Tx−dimt T + dimx X = dimw′Wv−dimwW + dimwWt+ dimv V −dimw′Wt

Since W → W is a smooth surjection, the same is true if we base-change over themorphism Specκ(v) → V (thinking of W → W as a morphism over V ), and fromthis smooth morphism we obtain the first of the following two equalities

dimwWv − dimw′Wv = dimw(Wv)w′ = dimwWw′ ;

the second equality follows via a direct comparison of the two fibres involved. Sim-ilarly, if we think of W → W as a morphism of schemes over T , and base-changeover some representative of the point t ∈ |T |, we obtain the equalities

dimwWt − dimw′Wt = dimw(Wt)w′ = dimwWw′ .

Putting everything together, we find that

dimt Tx − dimt T + dimx X = dimwWv − dimwW + dimv V.

Our goal is to show that the left-hand side of this equality vanishes for a non-emptyopen subset of t. As w varies over a non-empty open subset of W , its image t ∈ |T |varies over a non-empty open subset of |T | (as W → T is smooth).

We are therefore reduced to showing that if W → V is a scheme-theoreticallydominant morphism of irreducible locally Noetherian schemes that is locally offinite type, then there is a non-empty open subset of points w ∈ W such thatdimwWv = dimwW − dimv V (where v denotes the image of w in V ). This is astandard fact, whose proof we recall for the convenience of the reader.

We may replaceW and V by their underlying reduced subschemes without alteringthe validity (or not) of this equation, and thus we may assume that they are infact integral schemes. Since dimwWv is locally constant on W, replacing W bya non-empty open subset if necessary, we may assume that dimwWv is constant,say equal to d. Choosing this open subset to be affine, we may also assume thatthe morphism W → V is in fact of finite type. Replacing V by a non-empty opensubset if necessary (and then pulling back W over this open subset; the resultingpull-back is non-empty, since the flat base-change of a quasi-compact and scheme-theoretically dominant morphism remains scheme-theoretically dominant), we mayfurthermore assume that W is flat over V . The morphism W → V is thus ofrelative dimension d in the sense of Morphisms, Definition 29.1 and it follows fromMorphisms, Lemma 29.6 that dimw(W ) = dimv(V ) + d, as required.

Remark 5.23.0DS5 We note that in the context of the preceding lemma, it need notbe that dim T ≥ dimZ; this does not contradict the inequality in the statement ofthe lemma, because the fibres of the morphism f are again algebraic stacks, andso may have negative dimension. This is illustrated by taking k to be a field, andapplying the lemma to the morphism [Spec k/Gm]→ Spec k.

THE GEOMETRY OF ALGEBRAIC STACKS 21

If the morphism f in the statement of the lemma is assumed to be quasi-DM (in thesense of Morphisms of Stacks, Definition 4.1; e.g. morphisms that are representableby algebraic spaces are quasi-DM), then the fibres of the morphism over points ofthe target are quasi-DM algebraic stacks, and hence are of non-negative dimension.In this case, the lemma implies that indeed dim T ≥ dimZ. In fact, we obtain thefollowing more general result.

Lemma 5.24.0DS6 Let f : T → X be a locally of finite type morphism of Jacobson,pseudo-catenary, and locally Noetherian algebraic stacks which is quasi-DM, whosesource is irreducible and whose target is quasi-separated, and let Z → X denote thescheme-theoretic image of T . Then dimZ ≤ dim T , and furthermore, exactly oneof the following two conditions holds:

(1) for every finite type point t ∈ |T |, we have dimt(Tf(t)) > 0, in which casedimZ < dim T ; or

(2) T and Z are of the same dimension.

Proof. As was observed in the preceding remark, the dimension of a quasi-DMstack is always non-negative, from which we conclude that dimt Tf(t) ≥ 0 for allt ∈ |T |, with the equality

dimt Tf(t) = dimt T − dimf(t)Z

holding for a dense open subset of points t ∈ |T |.

6. The dimension of the local ring

0DS7 An algebraic stack doesn’t really have local rings in the usual sense, but we candefine the dimension of the local ring as follows.

Lemma 6.1.0DS8 Let X be a locally Noetherian algebraic stack. Let U → X be asmooth morphism and let u ∈ U . Then

dim(OU,u)− dim(ORu,e(u)) = 2 dim(OU,u)− dim(OR,e(u))

Here R = U ×X U with projections s, t : R→ U and diagonal e : U → R and Ru isthe fibre of s : R→ U over u.

Proof. This is true because s : OU,u → OR,e(u) is a flat local homomorphism ofNoetherian local rings and hence

dim(OR,e(u)) = dim(OU,u) + dim(ORu,e(u))

by Algebra, Lemma 111.7.

Lemma 6.2.0DS9 Let X be a locally Noetherian algebraic stack. Let x ∈ |X | be a finitetype point Morphisms of Stacks, Definition 18.2). Let d ∈ Z. The following areequivalent

(1) there exists a scheme U , a smooth morphism U → X , and a finite typepoint u ∈ U mapping to x such that 2 dim(OU,u)− dim(OR,e(u)) = d, and

(2) for any scheme U , a smooth morphism U → X , and finite type point u ∈ Umapping to x we have 2 dim(OU,u)− dim(OR,e(u)) = d.

Here R = U ×X U with projections s, t : R→ U and diagonal e : U → R and Ru isthe fibre of s : R→ U over u.

THE GEOMETRY OF ALGEBRAIC STACKS 22

Proof. Suppose we have two smooth neighbourhoods (U, u) and (U ′, u′) of x withu and u′ finite type points. After shrinking U and U ′ we may assume that u and u′are closed points (by definition of finite type points). Then we choose a surjectiveétale morphism W → U ×X U ′. Let Wu be the fibre of W → U over u and let Wu′

be the fibre of W → U ′ over u′. Since u and u′ map to the same point of |X | we seethat Wu ∩Wu′ is nonempty. Hence we may choose a closed point w ∈W mappingto both u and u′. This reduces us to the discussion in the next paragraph.

Assume (U ′, u′) → (U, u) is a smooth morphism of smooth neightbourhoods of xwith u and u′ closed points. Goal: prove the invariant defined for (U, u) is the sameas the invariant defined for (U ′, u′). To see this observe that OU,u → OU ′,u′ is aflat local homomorphism of Noetherian local rings and hence

dim(OU ′,u′) = dim(OU,u) + dim(OU ′u,u

′)

by Algebra, Lemma 111.7. (We omit working through all the steps to relate prop-erties of local rings and their strict henselizations, see More on Algebra, Section44). On the other hand we have

R′ = U ′ ×U,t R×s,U U ′

Thus we see that

dim(OR′,e(u′)) = dim(OR,e(u)) + dim(OU ′u×uU ′

u,(u′,u′))

To prove the lemma it suffices to show that

dim(OU ′u×uU ′

u,(u′,u′)) = 2 dim(OU ′u,u

′)

Observe that this isn’t always true (example: if U ′u is a curve and u′ is the genericpoint of this curve). However, we know that u′ is a closed point of the algebraicspace U ′u locally of finite type over u. In this case the equality holds because,first dim(u′,u′)(U ′u ×u U ′u) = 2 dimu′(U ′u) by Varieties, Lemma 20.5 and second theagreement of dimension with dimension of local rings in closed points of locallyalgebraic schemes, see Varieties, Lemma 20.3. We omit the translation of theseresults for schemes into the language of algebraic spaces.

Definition 6.3.0DSA Let X be a locally Noetherian algebraic stack. Let x ∈ |X | be afinite type point. The dimension of the local ring of X at x is d ∈ Z if the equivalentconditions of Lemma 6.2 are satisfied.

To be sure, this is motivated by Lemma 6.1 and Properties of Stacks, Definition 12.2.We close this section by establishing a formula allowing us to compute dimx(X ) interms of properties of the versal ring to X at x.

Lemma 6.4.0DSB Suppose that X is an algebraic stack, locally of finite type over alocally Noetherian scheme S. Let x0 : Spec(k) → X be a morphism where k is afield of finite type over S. Represent FX ,k,x0 as in Remark 2.11 by a cogroupoid(A,B, s, t, c) of Noetherian complete local S-algebras with residue field k. Then

the dimension of the local ring of X at x0 = 2 dimA− dimB

Proof. Let s ∈ S be the image of x0. If OS,s is a G-ring (a condition that is almostalways satisfied in practice), then we can prove the lemma as follows. By Lemma2.8, we may find a smooth morphism U → X , whose source is a scheme, containinga point u0 ∈ U of residue field k, such that induced morphism Spec(k) → U → X

THE GEOMETRY OF ALGEBRAIC STACKS 23

coincides with x0 and such that A = O∧U,u0. Write R = U ×X U . Then we may

identify O∧R,e(u0) with B. Hence the equality follows from the definitions.In the rest of this proof we explain how to prove the lemma in general, but we urgethe reader to skip this.First let us show that the right hand side is independent of the choice of (A,B, s, t, c).Namely, suppose that (A′, B′, s′, t′, c′) is a second choice. Since A and A′ are versalrings to X at x0, we can choose, after possibly switching A and A′, a formallysmooth map A → A′ compatible with the given versal formal objects ξ and ξ′

over A and A′. Recall that CΛ has coproducts and that these are given by com-pleted tensor product over Λ, see Formal Deformation Theory, Lemma 4.4. ThenB prorepresents the functor of isomorphisms between the two pushforwards of ξ toA⊗ΛA. Similarly for B′. We conclude that

B′ = B ⊗(A⊗ΛA) (A′⊗ΛA′)

It is straightforward to see thatA⊗ΛA −→ A⊗ΛA

′ −→ A′⊗ΛA′

is formally smooth of relative dimension equal to 2 times the relative dimensionof the formally smooth map A → A′. (This follows from general principles, butalso because in this particular case A′ is a power series ring over A in r variables.)Hence B → B′ is formally smooth of relative dimension 2(dim(A′) − dim(A)) asdesired.Next, let l/k be a finite extension. let xl,0 : Spec(l)→ X be the induced point. Weclaim that the right hand side of the formula is the same for x0 as it is for xl,0.This can be shown by choosing A→ A′ as in Lemma 2.5 and arguing exactly as inthe preceding paragraph. We omit the details.Finally, arguing as in the proof of Lemma 2.10 we can use the compatibilities inthe previous two paragraphs to reduce to the case (discussed in the first paragraph)where A is the complete local ring of U at u0 for some scheme smooth over X andfinite type point u0. Details omitted.

7. Other chapters

Preliminaries

(1) Introduction(2) Conventions(3) Set Theory(4) Categories(5) Topology(6) Sheaves on Spaces(7) Sites and Sheaves(8) Stacks(9) Fields(10) Commutative Algebra(11) Brauer Groups(12) Homological Algebra(13) Derived Categories

(14) Simplicial Methods(15) More on Algebra(16) Smoothing Ring Maps(17) Sheaves of Modules(18) Modules on Sites(19) Injectives(20) Cohomology of Sheaves(21) Cohomology on Sites(22) Differential Graded Algebra(23) Divided Power Algebra(24) Differential Graded Sheaves(25) Hypercoverings

Schemes

(26) Schemes

THE GEOMETRY OF ALGEBRAIC STACKS 24

(27) Constructions of Schemes(28) Properties of Schemes(29) Morphisms of Schemes(30) Cohomology of Schemes(31) Divisors(32) Limits of Schemes(33) Varieties(34) Topologies on Schemes(35) Descent(36) Derived Categories of Schemes(37) More on Morphisms(38) More on Flatness(39) Groupoid Schemes(40) More on Groupoid Schemes(41) Étale Morphisms of Schemes

Topics in Scheme Theory(42) Chow Homology(43) Intersection Theory(44) Picard Schemes of Curves(45) Weil Cohomology Theories(46) Adequate Modules(47) Dualizing Complexes(48) Duality for Schemes(49) Discriminants and Differents(50) de Rham Cohomology(51) Local Cohomology(52) Algebraic and Formal Geometry(53) Algebraic Curves(54) Resolution of Surfaces(55) Semistable Reduction(56) Derived Categories of Varieties(57) Fundamental Groups of Schemes(58) Étale Cohomology(59) Crystalline Cohomology(60) Pro-étale Cohomology(61) More Étale Cohomology(62) The Trace Formula

Algebraic Spaces(63) Algebraic Spaces(64) Properties of Algebraic Spaces(65) Morphisms of Algebraic Spaces(66) Decent Algebraic Spaces(67) Cohomology of Algebraic Spaces(68) Limits of Algebraic Spaces(69) Divisors on Algebraic Spaces(70) Algebraic Spaces over Fields(71) Topologies on Algebraic Spaces

(72) Descent and Algebraic Spaces(73) Derived Categories of Spaces(74) More on Morphisms of Spaces(75) Flatness on Algebraic Spaces(76) Groupoids in Algebraic Spaces(77) More on Groupoids in Spaces(78) Bootstrap(79) Pushouts of Algebraic Spaces

Topics in Geometry(80) Chow Groups of Spaces(81) Quotients of Groupoids(82) More on Cohomology of Spaces(83) Simplicial Spaces(84) Duality for Spaces(85) Formal Algebraic Spaces(86) Restricted Power Series(87) Resolution of Surfaces Revisited

Deformation Theory(88) Formal Deformation Theory(89) Deformation Theory(90) The Cotangent Complex(91) Deformation Problems

Algebraic Stacks(92) Algebraic Stacks(93) Examples of Stacks(94) Sheaves on Algebraic Stacks(95) Criteria for Representability(96) Artin’s Axioms(97) Quot and Hilbert Spaces(98) Properties of Algebraic Stacks(99) Morphisms of Algebraic Stacks(100) Limits of Algebraic Stacks(101) Cohomology of Algebraic Stacks(102) Derived Categories of Stacks(103) Introducing Algebraic Stacks(104) More on Morphisms of Stacks(105) The Geometry of Stacks

Topics in Moduli Theory(106) Moduli Stacks(107) Moduli of Curves

Miscellany(108) Examples(109) Exercises(110) Guide to Literature(111) Desirables(112) Coding Style(113) Obsolete

THE GEOMETRY OF ALGEBRAIC STACKS 25

(114) GNU Free Documentation Li-cense

(115) Auto Generated Index

References[EG17] Matthew Emerton and Toby Gee, Dimension theory and components of algebraic stacks.[Oss15] Brian Osserman, Relative dimension of morphisms and dimension for algebraic stacks,

J. Algebra 437 (2015), 52–78.